Equivalent condition for continuity at a point.

Question

I was studying the proof of the theorem that a bounded function $f:[a,b]\to\mathbb R$ is Riemann integrable iff it is continuous almost everywhere.In the proof $\alpha_P(x)=\sum\limits_{i=1}^n M_i \chi_{[x_{i-1},x_i)}$ and $\beta_P(x)=\sum\limits_{i=1}^n m_i \chi_{[x_{i-1},x_i)}$ where $M_i$ and $m_i$ are the suprema and infima in the interval $[x_{i-1},x_i)$ for the partition $P=\{a=x_0<x_1<...<x_n=b\}$.Then they consider a sequence of partitions $\{P_k\}$ such that $||P_k||\to 0$ and $P_{k+1}\supset P_k$ and define $\alpha_k=\alpha_{P_k}$ and $\beta_k=\beta_{P_k}$ and then they define $\alpha(x)=\lim\limits_{j\to \infty} \alpha_j(x)$ and $\beta(x)=\lim\limits_{j\to \infty} \beta_j(x)$.Now they claim that if $x\in [a,b]$ is not the end point of any interval defined by $P_k$ then $f$ is continuous if and only if $\alpha(x)=f(x)=\beta(x)$.I want to know two things.First of all,why is this so and secondly what happens for points which are end points of some $P_k$?Can someone answer my question?

Answer 1

It's easy to see that if $f$ is continuous at $x$ then $\alpha(x)=f(x)=\beta(x)$ Now, suppose $x$ is not an end point of any partition and $\alpha(x)=f(x)=\beta(x).$ So, we have two sequences $\{a_n\},\{b_n\}$ such that $a_n<b_n$ are consecutive points of partition $P_n$ and $x\in (a_n,b_n),b_n-a_n\mapsto 0.$ For,$y\in (a_n,b_n)$ then, $|f(y)-f(x)|\le M_{P_n}(a_n)-m_{P_n}(a_n)$ and $M_{P_n}(a_n)-m_{P_n}(a_n)\to 0$ as $n\to \infty.$ This answer your first question.

For 2nd question, consider the following function, $f(x)=1,x\ge 0$ and $-1,$ for $x<0$ and take the partition $P_=\{-1+\frac{1}{2^n},...,1\}$ ,now look at the point $x=0$ and check $f$ is not continuous there yet,it satisfies $f(0)=\alpha(0)=\beta(0).$